Math 411: Honours Complex Variables - University of Alberta

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$Math 527 A1 Homework 5 (Due Nov. 26 in Class) Proof. i. Set u = v(x ...$

74 CHAPTER 12. THE RESIDUE THEOREM AND APPLICATIONS Proof. Let ǫ > 0 be such that Bǫ(zj) ⊂ D for j = 1,...,n, with zk /∈ Bǫ(zj) for k �= j. For j = 1,...,n, we have Laurent representations f(z) = ∞� k=−∞ a (j) k k (z −zj) for z ∈ Bǫ(zj)\{zj}, so that res(f,zj) = a (j) −1. For j = 1,...,n, define hj: C\{zj} → C, z ↦→ so that hj is holomorphic on C\{zj}. Define �−1 k=−∞ g: D \{z1,...,zn} → C, z ↦→ f(z)− a (j) k (z −zj) k , n� hj(z), and note that z1,...,zn are removable singularities for g. Since D is simply connected, Cauchy’s Integral Theorem yields: � 0 = g(γ)dζ � = � = � = � = � = γ γ γ γ γ γ f(ζ)dζ − f(ζ)dζ − f(ζ)dζ − f(ζ)dζ − f(ζ)dζ − n� � j=1 n� � γ γ hj(ζ)dζ � −1 � j=1 k=−∞ n� �−1 a j=1 k=−∞ (j) k n� j=1 a (j) −1 � γ � j=1 a (j) k k (ζ −zj) γ 1 dζ ζ −zj (ζ −zj) k dζ n� res(f,zj)2πiν(γ,zj). j=1 Corollary 12.1.1. Let D ⊂ C be open and simply connected, f : D → C be holomorphic, and γ be a closed curve in D. Then we have ν(γ,z)f(z) = 1 � f(ζ) 2πi ζ −z dζ for z ∈ D\{γ}. γ � dζ
12.1. APPLICATIONS OF THE RESIDUE THEOREM TO REAL INTEGRALS75 Proof. Fix z ∈ D \{γ}, and define g: D \{z} → C, w ↦→ f(w) w−z . Then g is holomorphic with an isolated singularity at z. Let ∞� f(w) = an(w −z) n n=0 be the Taylor series expansion of f near z, so that ∞� g(w) = an+1(w −z) n , n=−1 and thus res(g,z) = a0 = f(z). The Residue Theorem then yields: � � f(ζ) 2πiν(γ,z)f(z) = 2πiν(γ,z) res(g,z) = g(ζ)dζ = ζ −z dζ. 12.1 Applications of the Residue Theorem to Real Integrals Proposition 12.1 (RationalTrigonometric Polynomials). Let p and q be polynomials of two real variables such that q(x,y) �= 0 for all (x,y) ∈ R 2 with x 2 +y 2 = 1. Then we have � 2π where 0 f(z) = 1 iz · p(cost,sint) q(cost,sint) � 1 p q dt = 2πi� res(f,z), γ z∈D � z + 2 1 � , z 1 � z − 2i 1 �� z � � 1 z + 2 1 � , z 1 � z − 2i 1 ��. z Proof. Just note that, by the Residue Theorem, 2πi � � res(f,z) = f(ζ)dζ z∈D = = ∂D � 2π f(e 0 iθ )ie iθ dθ � 2π 0 p(cosθ,sinθ) q(cosθ,sinθ) dθ. γ
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Math 411: Honours Complex Variables
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Contents 1 The Complex Numbers 5 2
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Chapter 1 The Complex Numbers Defin
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Proof. (i): If z = x+iy, then ¯z =
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Chapter 2 Complex Differentiation D
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(i) there exists c ∈ C such that
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Example. Let Then f is totally diff
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holds over C. We obtain: � ∞�
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Figure 3.2: Surface plot of cos(z)
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It follows that {an(z−z0) n } ∞
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Problem 3.1. Show in Theorem 3.2 th
Page 23 and 24: Definition. The length of a piecewi
Page 25 and 26: 1. Given a < b < c and two curves
Page 27 and 28: That is, F ′ (z) = f(z). Theorem
Page 29 and 30: ∆ (4) ∆ (1) ∆ (2) As the line
Page 31 and 32: Proof. Let z0 ∈ D be a center for
Page 33 and 34: four subtriangles, denoted by ∆(z
Page 35 and 36: γ1 Then it is clear that � � 1
Page 37 and 38: Proof. There is no loss of generali
Page 39 and 40: Proof. Apply Theorem 5.3 and 5.4. E
Page 41 and 42: is entire. Since p is a nonconstant
Page 43 and 44: (ii) for each compact K ⊂ D, the
Page 45 and 46: Proof. Let ζ ∈ ∂Br(z0), and no
Page 47 and 48: holds for some a0,a1,a2,... ∈ C a
Page 49 and 50: Proof. Let z0 ∈ D be such that |f
Page 51 and 52: Theorem 7.5 (Biholomorphisms of D).
Page 53 and 54: Define ⎧ ⎪⎨ g: D ∪{z0} →
Page 55 and 56: Define g: C → C, z ↦→ ∞�
Page 57 and 58: Chapter 9 Holomorphic Functions on
Page 59 and 60: It follows that � � f(ζ)dζ +
Page 61 and 62: Definition. The function h in Theor
Page 63 and 64: For the converse, let g: Br(z0) →
Page 65 and 66: Chapter 10 The Winding Number of a
Page 67 and 68: Proposition 10.2 (Winding Numbers A
Page 69 and 70: • if w �= z: � � |g(w,z)−
Page 71 and 72: Corollary 11.2.1. Let D be an open,
Page 73: Examples. 1. Let f(z) = eiz z 2 +1
Page 77 and 78: 12.1. APPLICATIONS OF THE RESIDUE T
Page 79 and 80: 12.1. APPLICATIONS OF THE RESIDUE T
Page 81 and 82: 12.2. THE GAMMA FUNCTION 81 12.2 Th
Page 83 and 84: 12.2. THE GAMMA FUNCTION 83 Since
Page 85 and 86: 12.2. THE GAMMA FUNCTION 85 6 5 4 3
Page 87 and 88: Chapter 13 Function Theoretic Conse
Page 89 and 90: Lemma 13.1. Let D ⊂ C be open and
Page 91 and 92: Definition. Let D ⊂ C be open, an
Page 93 and 94: Proof. Let p(z) = anz n +···+a1z
Page 95 and 96: for (x,y) ∈ R 2 \{(0,0)}. The par
Page 97 and 98: Proof. Of course, only (ii) =⇒ (i
Page 99 and 100: so that u(z) = � 2π u(re 0 iθ )
Page 101 and 102: Set K := supζ∈∂D|f(ζ)|. For z
Page 103 and 104: Chapter 15 Analytic Continuation al
Page 105 and 106: z0 ∈ D, the germ of f at z0—den
Page 107 and 108: Chapter 16 Montel’s Theorem Defin
Page 109 and 110: Define g: D → RM as follows: for
Page 111 and 112: and note that z−i 1+ z+i g(f(z))
Page 113 and 114: Example. Let z be a complex number.
Page 115 and 116: By the Open Mapping Theorem, there
Page 117 and 118: Proof. (i) =⇒ (ii) is Corollary 1
Page 119 and 120: Index C is a Field, 5 Biholomorphis
Page 121: INDEX 121 straight line segment, 25
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Math 411: Honours Complex Variables - University of Alberta

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